Interplanar spacings and intensities for these compounds are given in Table 11. Line intensities are given as peak height above background level of densitometer tracings from the film patterns; the most intense line is given a value of 1.00.
LITERATURE CITED
( 1 ) Bjerrum, J., McReynolds, J. P.,
“Inorganic Synthesis,” Vol. 11, p. 216, McGraw-Hill, New York, 1946. ( 2 ) Laitinen, H. A., Burdette, L. IT., ANAL.CHEM.23, 1268 (1951).
(3) Linhard, M., Flygare, H., 2. anorg. allgem. Chem. 262,328 (1950). (4)Linhard, M., Weigel, RI., Ibid., 263,245 (1950). ( 5 ) Lhhard, Lf., LF‘eigel, LI.,FlYW‘e, E., Ibid., 263, 233 (1950). RECEIVEDfor review June 25, 1957. Accepted September 10, 1957.
Quantitative X-Ray Diffraction Analysis L. E. COPELAND and ROBERT H. BRAGG’ Portland Cemenf Association, Research and Development laboratories, Chicago, 111.
b A general theory of quantitative x-ray analysis based on the theory of Alexander and Klug is derived and used to suggest several procedures for quantitative analysis. The need for calibration curves sometimes can b e eliminated; lines may be used even though there may be superpositions; several components may be determined simultaneously. The theory is applied to the determination of calcium hydroxide using magnesium hydroxide as the internal standard. Samples containing calcium hydroxide in the presence of hydrated calcium silicates have been analyzed. Precautions are necessary because of the carbonation of exposed samples.
Q
x-ray diffraction analysis usually consists of comparing the Debye-Scherrer pattern of a sample of the unknown, to which has been added a known amount of internal standard, with the patterns of standard mixtures whose compositions are known (8). These standard mixtures contain the substance sought and the internal standard in various proportions. X-ray intensity data from the standard mixtures are used for a calibration curve. Sometimes the strong lines of the x-ray pattern are not resolved and are, therefore, seemingly not suited for analysis. In this paper it is shown that the x-ray method can be applied to these problems by making a suitable application of the basic formula of Alexander and Klug UANTITATIVE
Unusual problems are encountered in dealing with hydrated silicates because most of the reaction products are extremely fine grained, having specific surfaces of the order of 100 square meters per gram, as measured by nitrogen adsorption. The calcium silicates are highly reactive to water vapor, and the calcium silicates, their hydrates, and calcium hydroxide are attacked by carbon dioxide in the laboratory atmosphere. Consequently, all operations must be carried out in a controlled atmosphere. GENERAL THEORY
196
ANALYTICAL CHEMISTRY
Equation 1 refers to integrated intensity data excluding the general background radiation; the use of maximum or peak intensities may lead to gross errors in analyses. When p$ is not known (the usual case), Equation 1 can be written in the form
Consider Bragg and Bragg (4) reflections from the surface of a flat powder sample such as is used in the Norelco diffractometer. The sample is a uniform mixture of particles small enough to make extinction and microabsorption effects negligible (6, 16), and its thickness is such that the diffracted x-rays have maximum intensity. Sample porosity must be constant in all comparisons of intensities; otherwise the intensity of x-ray reflections will vary. In addition, the crystallites in the sample must be oriented randomly. Alexander and Klug (1) have shown that for samples meeting these requirements, the basic equation relating the intensity of the diffracted x-rays to the absorptive properties of the sample is
where p> is the total mass absorption coefficient of the sample including component j . The quantity p $ can be measured experimentally, and because Ki, can be determined by measurements on pure component j , and p, will be known, analyses based on Equation 2 are feasible. However, such analyses have been shown to require empirical corrections, probably because of microabsorption effects (18, 16). For another component I, of the mixture, an equation can be written for the kth line in the pattern of this substance
where It1
wherein the values of the quantities involving the indices k, I may differ from those having indices i,j . Because Idi and Ihl are measured on the same sample, and p $ is characteristic of the sample, the ratio of these intensities is
(1).
The theory is applied t o the quantitative determination of calcium hydroxide in the presence of hydrated calcium silicates. In the hydration of cement or of calcium silicates, calcium hydroxide is liberated; thus the method should be useful when studying the kinetics or stoichiometry of such reactions. Results sufficiently accurate for most purposes can be obtained within 3 hours. Present address, Armour Research Foundation, Chicago, Ill.
K1.5 =
terial which would remain if component j were removed) constant which depends on the characteristics of the apparatus and on the structure of component j
25
p5
WT p:
intensity of the ith line in the Debye-Scherrer pattern of component j whose concentration is sought = concentration of component j in grams per gram of sample = density of componentj = mass absorption coefficient of component j = mass absorption coefficient of the matrix (the ma=
(3)
Equations 1 to 4 were first derived by Alexander and Klug (1). Equation 4 is the formula used by numerous investigators ( I , 5, 7-9, 11, 15) and is
fundamental to the internal standard method of quantitative x-ray diffraction analysis. It eliminates the small errors resulting from microabsorption effects, A graph of I,,/Ikl plotted us. x,/xl, the calibration curve, will be a straight line passing through the origin and . slope having slope ( K I j p 1 / K k l p , )The is obtained from x-ray intensity measurements on mixtures for which x , / x l is known. Dilution of Sample. T o determine the amount of unknown in a sampleLe., component j-a knon n amount of the internal standard, component 1, is added. The intensity ratio, I{j/Ikl, is measured, and the value of 5 , is computed by substituting the known quantities in Equation 4. This is the internal standard method most frequently described in the literature. Instead of using an internal standard the sample can be diluted with known amounts of the unknown, component j. The sample already contains an amount 21 of a second component, 1. E , grams of component j per gram of original sample is added. The weight fraction of component j is now ___ that of
xi
+
(constant) (xi
+
Ej)
+
Iij
Zkl
=
CU,Zll
+
(8)
PiJll
Dividing Equation 8 by I l l , the intensity of a well resolved line in the pattern of the internal standard gives
Zli _ = -K I P~I zj Kit
111
p j 51
Consequently,
where I
16.0
17.0 18.0 BRAGG ANGLE, ¶e
19.0
KxPi and Pk are constants.
ai K1j
Thus, if the two lines are not resolved but both included in one measurement a graph of "j + Ik' plotted us. will be TI 1 a straight line. Equation 11 is independent of the degree of superposibion. It would be valid even if the lines were resolved completely, or if one m-as superimposed totally upon t'he other. Previous investigators avoided superpositions (16). They present no unsolvable problems if the components from which they arise can be identified; then the relative intensities can be determined on the pure compounds. With a little algebraic manipulation of Equations 11 and 7 formulas can be obtained for unscrambling total intensity measurements-Le., from measI k zand Ilj,I l l can be urements of I,, deduced. Simultaneous Analysis for Several Components. The methods described above, when suitably extended, can be used for estimating any number of components of a mixture. If bhe number to be estimabed is n,then a t least n intensities mcst be measured for each sample, regardless of the degree to which lines from different components are superimposed. If the mth intensity is I,, then
2
~
Figure 1. Partial superposition of 001 reflections of calcium hydroxide and magnesium hydroxide
(5)
In this case, a graph of I , , / I k Zplotted as a function of E , will be a straight line, but it will not pass through the origin. The vertical intercept a t E, = 0 is Itj/Ikl for the original undiluted sample. .4t Isj/Ikl = 0 , (2,
